{"id":"f2906bb8-8798-4b7d-8241-f9f7386e63d2","arxiv_id":"2502.04464","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Integer-ratio rhythm tests that normalize counts by bin width implicitly test against a Poisson-process null; the paper provides explicit alternative nulls and demonstrates their effect on birdsong and music data.","lead":"This paper shows that the standard method for testing integer-ratio rhythms in animal and human music, which counts ratios in bins and divides by bin width, silently assumes the timings come from a random Poisson process. It gives researchers formulas and code to test against any other null hypothesis, and demonstrates on four real datasets that conclusions can change when the null is chosen differently.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'de facto Poisson null' claim is stronger than the math supports: bin-width normalization directly tests only local flatness of r_k density, and the Poisson identification requires iid exponential intervals plus independence of the r_k sample, both of which real sequences violate.","rationale":"I read the paper in good faith. The mathematical core is sound: for independent exponentially distributed intervals, r_k = i_k/(i_k+i_{k+1}) is uniform on [0,1], so bin-width normalization of counts is calibrated to a uniform r_k distribution. The worked reanalysis in Section 7 and the code in the Supporting Information are useful and reproducible, and the general recommendation to make null hypotheses explicit is well taken. The load-bearing weakness is the step from 'uniform r_k marginals' to 'the implicit null hypothesis is a Poisson point process.' That step requires the empirical intervals to be statistically independent draws from a single exponential distribution, and it requires the sequence of r_k values used in the test to behave like an iid sample. Neither requirement is stated or justified in the manuscript. In actual recordings, adjacent intervals are often dependent, and the overlapping construction of r_k induces dependence among successive ratios even when intervals are independent. The bootstrap procedure compounds this by resampling r_k values with replacement as if they were exchangeable. Consequently, the strong attribution that previous studies 'de facto' tested a Poisson null is an overstatement. This does not undermine the paper's central methodological warning, because the low-bar problem and the inflation of 1:1 ratios under bounded interval distributions follow from the marginal distribution of r_k alone. The recommended change is a caveat in Sections 3, 5, and 8 distinguishing the directly tested uniform-r_k null from the Poisson-process interpretation. Since the reader's verdict was already CONDITIONAL and this concern is addressable without changing the overall assessment, I leave the verdict unchanged.","tokens_in":25629,"tokens_out":5819,"duration_ms":69173,"concrete_test":"Generate 10,000 stationary sequences of length n=1000 from a Gaussian-copula AR(1) process with exponential marginals (or a two-state Markov-switching exponential process with different mean durations), so every marginal interval is Exp(1) but intervals are serially correlated. Apply the exact normalized-bin bootstrap comparison used in Section 7.3 for the 1:1 ratio, with nominal alpha=0.05. Record the proportion of sequences in which the on-1:1 and off-1:1 bins are declared significantly different. If this proportion is close to 5%, the Poisson identification is robust to dependence; if it is materially larger (e.g., >10%), the 'de facto Poisson' claim needs qualification, since a non-Poisson dependent process with exponential marginals triggers the same test.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5's central assertion that bin-width normalization 'de facto tests the implicit null hypothesis that the (binned) empirical rhythm ratios have been generated by a Poisson process' is stronger than what Sections 4.2-4.3 establish. Equation 7 gives the density of r for two independent draws from an interval density p_I, and Equation 10 shows that for p_I exponential this density is uniform. That proves a Poisson process is a sufficient condition for uniform r_k marginals; it does not prove that uniform r_k marginals imply a Poisson process. Real r_k sequences use overlapping pairs, since i_k is shared by r_{k-1} and r_k, so even under a Poisson process the binned counts are not iid draws. More importantly, serial dependence in the interval-generating process can change the joint null distribution of the test statistic while leaving each marginal r_k uniform. The bootstrap in Section 7.3 resamples r_k values with replacement, implicitly treating them as exchangeable, an assumption violated by overlapping ratios. Thus the claim that 'almost all of past research' tested a Poisson null relies on an unstated uniqueness and independence premise. The paper's practical warning, that bounded or otherwise non-exponential interval distributions can produce spurious 1:1 peaks in normalized bin counts, is independent of this premise and remains well supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper examines the statistical assumptions underlying integer-ratio analyses of temporal sequences in bioacoustics and music, focusing on the rhythm ratio r_k = i_k/(i_k+i_{k+1}). It derives the density of r_k for arbitrary interval distributions (Eq. 7), shows that exponentially distributed intervals (a Poisson process) yield a uniform r_k distribution (Eq. 10), and argues that the common practice of normalizing bin counts by bin width implicitly tests a uniform null, which the authors equate with a Poisson null. The paper further derives normalization factors for alternative null hypotheses (uniform and log-normal interval distributions) and reanalyzes four datasets from Roeske et al. (2020) to illustrate how the choice of null changes the statistical conclusions. The takeaway message is that researchers should make their null hypothesis explicit.","tokens_in":25900,"tokens_out":11650,"duration_ms":113363,"significance":"The paper addresses a real and timely problem: the popular method for detecting integer-ratio rhythms has not had its null hypothesis carefully examined. The central derivations are correct and clearly presented; Eq. (7) is a standard change of variables, and Eq. (10) correctly shows that the r_k ratio of independent exponential intervals is uniform. The proposed normalization formula (Eq. 16) and the worked examples provide a practical toolkit for researchers. The paper also makes a falsifiable point—that bounded or log-normal interval distributions can create spurious 1:1 peaks—which is independent of the stronger Poisson-process claim. However, as detailed below, the interpretation of the implicit null as 'de facto Poisson' goes beyond what the mathematics establishes, and the reanalysis has some methodological weaknesses. Overall, the core contribution is valuable, but the framing needs revision.","major_comments":[{"comment":"The statement that bin-width normalization 'de facto tests the implicit null hypothesis that the (binned) empirical rhythm ratios have been generated by a Poisson process' is stronger than the derivations support. Equations (7) and (10) show that if the intervals i_k are independent and exponentially distributed, then r_k is uniform; they do not show the converse, that a uniform r_k distribution implies a Poisson process. The test based on normalized bin counts directly tests uniformity of the r_k values. A Poisson process is a sufficient but not necessary condition for that uniformity, so the claim that 'almost all of past research' has tested a Poisson null is not justified. Additionally, the r_k sequence from adjacent intervals is not independent (r_k and r_{k+1} share i_{k+1}), so even under a Poisson process the joint distribution of the binned counts is not that of iid draws; the paper should either present a rigorous argument for the equivalence or rephrase the claim as 'the null hypothesis is a uniform distribution of r_k, which is the distribution generated by a Poisson process.'","section":"Section 5 (paragraph 2) and Section 8"},{"comment":"The reanalysis fits the uniform bounds (a,b) to the observed min/max durations and the log-normal parameters (µ,σ) to the observed intervals in the same datasets used for the subsequent tests. Fitting null-distribution parameters to the test data can make the null artificially close to the data, biasing the comparison towards non-significance. The paper should either fix the null parameters a priori (e.g., from theoretical constraints) or account for parameter estimation in the null distribution. Furthermore, the bootstrap resamples r_k values with replacement, treating them as exchangeable, but overlapping ratios induce dependence; this affects the confidence intervals. The authors should acknowledge these limitations or use a resampling scheme that preserves dependence (e.g., resampling intervals).","section":"Section 7.3, null-fitting and bootstrap"}],"minor_comments":[{"comment":"Pervasive empty citation placeholders '()' appear throughout (e.g., Section 1, paragraphs 1-2; Section 2, paragraph after equation (1); Section 3, paragraph 5; Section 5, paragraph 2; Section 6.3, last paragraph before Eq. (16); Section 7.3, paragraphs 1-2). These must be completed before the manuscript can be evaluated.","section":"Throughout"},{"comment":"The caption contains 'a a trumpet'; this should be 'a trumpet'.","section":"Figure 1 caption"},{"comment":"In the first paragraph, 'statically' should be 'statistically'.","section":"Section 8"},{"comment":"The caption says 'Cuban salsa finch dataset'; this should be 'Cuban salsa dataset'.","section":"Table S3 caption"},{"comment":"The phrasing 'we can deduce the exact ratio distribution' is informal; consider 'we now derive'.","section":"Section 7.1"},{"comment":"The notation for bin edges, e.g., '0.444...' for 4/9, is unconventional; using fractions such as 4/9 would improve clarity.","section":"Section 7 (bins)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a large number of empty citation placeholders, suggesting an incomplete draft; the editor should require a complete reference list before review. The paper's scope matches the journal, but the central interpretive claim needs the revisions outlined above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the paper the integer-ratio subfield needed. The central observation—that normalizing bin counts by width in the r_k method implicitly assumes a uniform distribution of ratios, and that under i.i.d. exponential intervals this is the signature of a Poisson process—is mathematically correct (Eq. 7 is a textbook change of variables; Eq. 10 follows) and practically important. The framework in Sections 6 and 7 for testing alternative nulls (uniform, log-normal intervals) via rescaling or adjusted normalization factors is new in this literature and genuinely useful. The reanalysis of the Roeske et al. datasets shows that the choice of null changes which species or music appear to have categorical rhythms; the zebra finch 1:1 effect disappears under a uniform-interval null. That is a substantive result.\n\nThe paper is less careful about the necessity direction. Uniform r_k marginals are a necessary consequence of a Poisson process, but the normalized bin-count test only tests uniformity of r_k. Other interval distributions can also produce non-uniform r_k, and there may be non-Poisson processes with uniform r_k. So saying the test 'de facto tests the null that the data were generated by a Poisson process' is too strong. The paper itself acknowledges the practical side of this (bounded interval distributions produce spurious 1:1 peaks), but the wording in Section 5 and the conclusion should be softened to 'implicitly assumes a uniform ratio distribution, which is the Poisson-process benchmark.' The second soft spot is the independence premise: the derivations assume i.i.d. interval draws, while real sequences have overlapping ratios and temporal dependencies. The bootstrap in Section 7.3 treats the r_k as exchangeable, which is dubious. This does not break the paper's main argument, but it should be acknowledged. Finally, the reanalysis fits the log-normal null to the same data and applies a 1–99 percentile outlier exclusion, so those results are illustrative rather than evidential.\n\nThe math checks out, the code is included, and the paper is clearly written. It deserves a serious referee; the overclaims are fixable with a moderate revision.","headline":"A correct and useful diagnosis of the implicit Poisson null in rhythm-ratio binning, slightly overstated in its necessity claim but well worth refereeing.","tokens_in":26444,"tokens_out":2877,"would_cite":true,"duration_ms":29500,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Standard integer-ratio rhythm tests secretly assume a Poisson process generates the intervals.","keywords":["integer ratios","rhythm analysis","Poisson process","null hypothesis","bin-width normalization","bioacoustics","music cognition","interval distributions"],"falsifier":"Simulate surrogate interval sequences that preserve the observed marginal distribution and autocorrelation but contain no deliberate integer-ratio structure, then run the width-normalized bin test; if these surrogates reject more than the nominal rate, the Poisson-null interpretation of the test does not describe real data.","tokens_in":25435,"feed_emoji":"🎵","tokens_out":8039,"duration_ms":77272,"temperature":0.7,"pith_summary":"This paper argues that the standard way of testing for small-integer rhythmic ratios in animal calls and music, which counts how many adjacent-interval ratios fall into bins around ratios such as 1:1 and 2:1 and divides each count by the bin's width, silently assumes a particular null model: that the intervals were generated by a Poisson process, so that rhythm ratios are uniformly distributed. Because a Poisson process is an extremely permissive, maximally random baseline with no bound on interval duration, positive results from past studies are weaker evidence for categorical rhythm than they appear. The authors derive the exact probability distribution of the rhythm ratio under any chosen interval distribution and give two equivalent fixes: rescale the ratio so that the chosen null looks uniform, or replace bin width with the null's expected probability mass per bin. Reanalyzing birdsong and music datasets, they show that some previously significant 1:1 findings, such as the zebra finch result, disappear under a uniform-interval null, while other ratio peaks persist.","feed_headline":"Integer-ratio rhythm tests silently assume Poisson timing","feed_subtitle":"Standard bin-width normalization secretly compares songs against random timing; the paper shows how to pick a fairer null.","key_machinery":"The load-bearing object is the rhythm ratio $r_k = i_k/(i_k+i_{k+1})$, which maps any pair of adjacent intervals to $[0,1]$ and is invariant to tempo. The mathematical identity that carries the argument is the change-of-variables formula giving the density of $r_k$ from the interval density $p_I$, together with the fact that $r_k$ is uniform when $p_I$ is exponential. The paper also derives the normalization constant $\\hat{w}_{I,u,v} = \\int_u^v p_S(s) ds$ that replaces bin width when the chosen null is not uniform.","core_discovery":"The paper's central claim is that when empirical ratios $r_k = i_k/(i_k+i_{k+1})$ are binned and each bin count is divided by its width, the test compares the data against a uniform distribution of ratios on $[0,1]$, and that uniform distribution is exactly what a homogeneous Poisson point process produces for exponentially distributed intervals. The authors therefore state that most prior integer-ratio analyses have de facto tested the null hypothesis that the binned rhythm ratios were generated by a Poisson process, a maximally random baseline with no upper or lower bound on interval duration. They then show that any scale-invariant ratio formula must depend only on the interval fraction $q = i_2/i_1$, derive the general density $p_R(r)=r^{-2} \\int_0^\\infty t p_I(t) p_I(t(1-r)/r) dt$, and provide two equivalent ways to test an arbitrary null distribution: transform the ratio so that the null becomes uniform, or normalize bin counts by the null's expected probability mass inside each bin. Reapplying the method to thrush nightingale, zebra finch, Cuban salsa, and Malian jembe data, they report that some 1:1 peaks, such as the zebra finch result, cease to be significant under a uniform-interval null, while other ratio peaks survive.","pith_inferences":["The same bin-width critique should apply to any analysis that bins ratios of adjacent intervals and normalizes by width—for example gait kinematics or speech timing—so the reweighting recipe may be directly portable to those fields.","A natural testable extension is to estimate the interval distribution per species or piece from data and compare integer-ratio models against that empirical null using likelihood or goodness-of-fit, which would make the null choice data-driven rather than assumed.","Because any scale-invariant ratio discards tempo, the framework says nothing about tempo drift within a performance; studying tempo drift would require a separate statistic that keeps the absolute scale.","If the independence assumption fails, a bootstrap that resamples individual $r_k$ values will not reflect the true null; a block or pair-resampling scheme would be more appropriate."],"forward_implications":["Results of past integer-ratio studies should be reread as evidence against a Poisson process, not against rhythm-free behavior in general.","Future empirical work should state an explicit null distribution and, when it is not uniform, normalize bin counts by expected probability mass under that null.","The rhythm ratio $r_k$ remains a defensible default because it is scale-invariant, bounded, symmetric around 1:1, and maps Poisson-generated intervals to a uniform distribution.","Choosing a different ratio formula or reweighting the same $r_k$ can change which integer-ratio peaks reach significance, as the zebra finch 1:1 example shows.","For nulls such as uniform or log-normal intervals, the paper supplies closed-form densities and normalization constants, and for other nulls a Monte Carlo approximation."],"supporting_citations":[{"why":"Defines the rhythm ratio $r_k$, the binning and width-normalization procedure, and the birdsong and music datasets that the paper reanalyzes.","marker":"Roeske et al. (2020)"},{"why":"Supplies the cross-cultural evidence for integer-ratio priors that motivates the search for categorical rhythm in empirical sequences.","marker":"Jacoby & McDermott (2017)"},{"why":"Provides the thrush nightingale and zebra finch interval datasets used in the worked reanalysis.","marker":"Roeske & Tchernichovski (2020)"},{"why":"Provides the Cuban salsa and Malian jembe interval datasets used in the worked reanalysis.","marker":"Roeske & Jacoby (2020)"},{"why":"Underlies the use of the exponential distribution and Poisson process as the canonical random-timing baseline.","marker":"Ross (2019)"},{"why":"Supplies the change-of-variables results used to derive the probability density of rhythm ratios.","marker":"Springer (1979)"}],"fun_headline_variants":["Integer-ratio rhythm tests hide a Poisson timing assumption","Poisson null silently skews music rhythm ratio findings","A fairer null for integer-ratio rhythm research","Why past rhythm 1:1 peaks may be Poisson artifacts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The identification of bin-width normalization with a Poisson null holds only if the intervals are independent draws from a single exponential distribution; real recordings have temporal dependencies and overlapping adjacent ratios, so the effective null may differ.","fun_headline_variants_meta":{"raw":{"variants":["Integer-ratio rhythm tests hide a Poisson timing assumption","Poisson null silently skews music rhythm ratio findings","A fairer null for integer-ratio rhythm research","Why past rhythm 1:1 peaks may be Poisson artifacts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000441,"raw_usage":{"total_tokens":2274,"prompt_tokens":1023,"completion_tokens":1251,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":1186}},"tokens_in":639,"tokens_out":1251,"duration_ms":14107,"temperature":1.0,"reasoning_tokens":1186,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:36:50.879698+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate surrogate interval sequences that preserve the observed marginal distribution and autocorrelation but contain no deliberate integer-ratio structure, then run the width-normalized bin test; if these surrogates reject more than the nominal rate, the Poisson-null interpretation of the test does not describe real data.","supporting_citations":[{"cited_title":"C., Tchernichovski, O., Poeppel, D., & Jacoby, N","cited_arxiv_id":null,"evidence_quote":"Defines the rhythm ratio $r_k$, the binning and width-normalization procedure, and the birdsong and music datasets that the paper reanalyzes."},{"cited_title":"C., Tchernichovski, O., Poeppel, D., & Jacoby, N","cited_arxiv_id":null,"evidence_quote":"Provides the thrush nightingale and zebra finch interval datasets used in the worked reanalysis."},{"cited_title":"C., Tchernichovski, O., Poeppel, D., & Jacoby, N","cited_arxiv_id":null,"evidence_quote":"Provides the Cuban salsa and Malian jembe interval datasets used in the worked reanalysis."}],"review_version":1}